Determining whether the series is convergent or divergent

In summary, a convergent series is one in which the sum of its terms approaches a finite value. To determine whether a series is convergent or divergent, various tests such as the limit comparison test or integral test can be used. Absolute convergence refers to a series in which the sum of the absolute values of its terms converges, while conditional convergence refers to a series in which the sum of its terms converges but not the sum of the absolute values of its terms. A series cannot be both convergent and divergent, but it can be conditionally convergent. There are many real-life applications of determining whether a series is convergent or divergent, such as in finance, physics, and engineering.
  • #1
umzung
21
0

Homework Statement


Determine if the series is convergent.

Homework Equations



∑ (((2n^2 + 1)^2)*4^n)/(2(n!))
n=1[/B]

The Attempt at a Solution


I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the limit to be < 1 to show this. Is there way to now divide each term by the dominant term n^2, or do I need to multiply the whole thing out and divide by the new dominant term? I've tried that and have found the limit to be 0/4 = 0.
 
Physics news on Phys.org
  • #2
Just note that the numerator goes as ##8n^4 + \mathscr O(n^3)## and that the denominator goes as ##4n^5 + \mathscr O(n^4)## as ##n\to \infty##. This will be sufficient.
 
  • #3
umzung said:

Homework Statement


Determine if the series is convergent.

Homework Equations



∑ (((2n^2 + 1)^2)*4^n)/(2(n!))
n=1[/B]

The Attempt at a Solution


I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the limit to be < 1 to show this. Is there way to now divide each term by the dominant term n^2, or do I need to multiply the whole thing out and divide by the new dominant term? I've tried that and have found the limit to be 0/4 = 0.
If you want to show the algebra, you can do something like this:
$$\frac{4}{n+1}\frac{[2(n+1)^2+1]^2}{(2n^2+1)^2}
=\frac{4}{n+1}\frac{\left[(n+1)^2\left(2+\frac{1}{(n+1)^2}\right)\right]^2}{\left[n^2\left(2+\frac{1}{n^2}\right)\right]^2}$$ so you don't have to multiply everything out.
 

1. What is the definition of a convergent series?

A convergent series is a series in which the sum of its terms approaches a finite value as the number of terms increases. In other words, the terms in a convergent series become smaller and smaller, eventually approaching zero.

2. How do you determine whether a series is convergent or divergent?

To determine whether a series is convergent or divergent, you can use several different tests such as the limit comparison test, ratio test, or integral test. Each test has its own set of criteria and can be used in different scenarios. Ultimately, the goal is to show that the series either approaches a finite value (convergent) or increases without bound (divergent).

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series in which the sum of the absolute values of its terms converges. Conditional convergence, on the other hand, refers to a series in which the sum of its terms converges, but the sum of the absolute values of its terms does not. In other words, a series can be conditionally convergent, but not absolutely convergent.

4. Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. By definition, a convergent series approaches a finite value, while a divergent series does not. However, a series can be conditionally convergent, as mentioned in the previous question.

5. Are there any real-life applications of determining whether a series is convergent or divergent?

Yes, there are many real-life applications of determining whether a series is convergent or divergent. For example, in finance and economics, series can be used to model interest rates, economic growth, and stock prices. In physics, series can be used to model the behavior of particles and waves. In engineering, series can be used to analyze electrical circuits and signal processing. In all of these applications, determining whether a series is convergent or divergent is crucial in making accurate predictions and calculations.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
314
  • Calculus and Beyond Homework Help
Replies
5
Views
928
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
640
  • Calculus and Beyond Homework Help
Replies
4
Views
855
  • Calculus and Beyond Homework Help
Replies
2
Views
645
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
722
  • Calculus and Beyond Homework Help
Replies
7
Views
908
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
Back
Top